Surface Di®usion Oleg M. Braun Institute of Physics National Academy of Sciences of Ukraine Contents 1 Introduction 3 2 Di®usion of a Single Atom 5 2.1 Di®usion: what is it? . 5 2.2 Langevin and Fokker-Planck-Kramers equations . 7 2.3 Solution: one-dimensional system . 10 2.3.1 Conventional di®usion equation . 10 2.3.2 The FPK equation in the absence of the substrate potential . 11 2.3.3 General case . 12 2.3.4 Smoluchowski equation . 13 2.3.5 Low-friction limit . 17 2.3.6 Intermediate friction . 19 2.4 Kramers theory . 20 2.5 Computer simulation: Molecular Dynamics . 23 2.6 Lattice models . 24 3 Collective Di®usion 27 3.1 Mechanisms of interaction between adsorbed atoms . 27 3.2 Di®erent di®usion coe±cients . 28 3.2.1 Susceptibility . 28 3.2.2 Self-di®usion coe±cient . 29 3.2.3 Mobility . 30 3.2.4 Chemical di®usivity . 30 3.3 Ideology of quasiparticles . 32 3.4 One-dimensional chain (the Frenkel-Kontorova model) . 32 3.5 Gas model . 32 3.6 Lattice models . 32 3.7 2D Frenkel-Kontorova model . 33 3.8 Phase transitions . 33 3.8.1 First-order phase transition . 33 3.8.2 Second-order phase transition . 33 3.9 Computer simulation . 34 3.9.1 Monte Carlo simulation . 34 3.9.2 Molecular Dynamics simulation . 34 4 Defects 35 4.1 ... 35 1 2 CONTENTS 5 Experimental Methods 37 6 Results 39 6.1 Quasi-one-dimensional systems . 39 6.2 ... 39 7 Conclusion 41 References 42 Subject Index 43 Chapter 1 Introduction Surface di®usion deals with very interesting physics, namely: ² surface di®usion may be considered as motion in an external (substrate) potential; thus, here we meet with low-dimensional (two-dimensional) physics, or even with one-dimensional situation as for di®usion in channels of furrowed surfaces like W(112); ² \transparent" { atomic motion on a plane may easily be visualized; surface di®usion may be studied by direct experimental methods (although experimental techniques are not too simple { a surface should be specially prepared to have a well-de¯nes structure without defects or with controlled defects, one should use high vacuum to avoid impurities, the technique should be with high spatial resolution, one should take care to prevent evapo- ration or absorption into the bulk); ² explores the very wide concentration interval from θ = 0 to 1 or even more. Surface di®usion is very important practically, namely: ² growing of crystals; ² emission electronics; ² catalysis; ² corrosion; ² soldering, welding, powder metallurgy. Theoretically the problem reduces to motion of atoms in an external (periodic) potential when the interaction between the atoms must be taken into account. What will not be described in this book: di®usion on disordered lattices (see excellent survey of Haus and Kehr (1987)). However, a role of defects will be considered briefly. 3 Chapter 2 Di®usion of a Single Atom 2.1 Di®usion: what is it? Classical di®usion (or Brownian motion, or random walk) is de¯ned as such a motion of a particle that its mean-square displacement from an initial point is proportional to time, h(r(t) ¡ r(0))2i = 2ºDt ; t ! 1 : (2.1) Here h:::i stands for the averaging over the equilibrium state of the system, and º = 1 for the 1D system, º = 2 for the 2D (surface) di®usion, and º = 3 for the 3D (bulk) di®usion. Using Eq. (2.1), the di®usion coe±cient D may be de¯ned as ¿ À ¿Z À 1 d 1 1 t D = lim (r(t) ¡ r(0))2 = lim hv(t)(r(t) ¡ r(0))i = lim dt0 v(t)v(t0) ; (2.2) 2º t!1 dt º t!1 º t!1 so that Z 1 1 D = d¿ hv(0)v(¿)i: (2.3) º 0 It is natural to generalize this de¯nition and to introduce the tensor di®usion coe±cient, Z 1 i!¿ D®¯(!) = d¿ e hv®(0)v¯(¿)i; (2.4) 0 1 Pº so that D = º ®=1 D®®(0). To use Eqs. (2.3) or (2.4), ¯rst we have to know a corresponding motion equation as well as its solution. In statistical mechanics the system state is described by the distribution function f(r; p; t), which depends on the coordinate r and the momentum p = mv of a particle of mass m, and in the absence of a time-dependent external perturbation the system must approach to the equilibrium state, f(r; p; t ! 1) ! f0(r; p), which corresponds to the Maxwell-Boltzmann distribution µ ¶º ½ · ¸¾ 1 1 1 2 f0(r; p) = N p exp ¡ p + '(r) ; (2.5) 2¼mkBT kBT 2m where '(r) is the potential energy the particle; the function '(r) should be 2D-periodic along the surface. N is an appropriate normalization; usually it is convenient to normalize on one particle per unit of volume (or area in the case of surface di®usion, or length for the 1D di®usion). For example, for the 1D potential with the period as the normalization factor is Z µ ¶ 1 as '(x) N ¡1 = dx exp ¡ : (2.6) as 0 kBT 5 6 CHAPTER 2. DIFFUSION OF A SINGLE ATOM Evolution of the distribution function may often be described by the equation f_ = Lf: (2.7) Di®erent methods of deduction of Eq. (2.7) are described, e.g., in []. (Zaslavsk) For a Hamiltonian system, the operator L corresponds to the Liuville operator, µ ¶ @ @ X @ @ LH = ¡v ¡ p_ = ¡ v® + F® ; (2.8) @x @p ® @x® @p® where F® = ¡@'(r)=@x® is the force acting on the particle. The motion equation allows us to couple the di®usion coe±cient D with the mobility coe±- cient B. The latter describes a linear response of the system being in the equilibrium state, to an action of in¯nitesimal external force ±F (x; t). Namely, the external perturbation leads to the change of the distribution function on ¢f = f ¡ f0 ; j¢fj ¿ jf0j : (2.9) According to (2.7), the deviation ¢f should satisfy the linearized equation (Lf0 = 0) @ (¢f) = (L + ¢L)(f + ¢f) ¼ L ¢f + ¢L f ; (2.10) @t 0 0 a solution of which may formally be written in the form Z t 0 £ 0 ¤ 0 ¢f(t) = dt exp L(t ¡ t ) ±L(t ) f0 : (2.11) A deviation of the distribution function from the equilibrium one leads to appearance of the flux of particles with the density Z p hj(x; t)i = dp ¢f(x; p; t) : (2.12) m @ Substituting (2.11) into (2.12) and using ±L(t) = ¡±F (x; t) @p which follows from Eq. (2.8), and incorporating also the equation @f0=@p = ¡f0v=kBT which follows from (2.5), we obtain Z t ½Z ¾ ¡1 0 L(t¡t0) 0 hj®(x; t)i = (kBT ) dt dp f0(x; v) v®e v¯ ±F¯(x; t ) : (2.13) De¯ning now the mobility tensor by the relation Z t 0 0 0 hj®(x; t)i = dt B®¯(t ¡ t ) ±F¯(x; t ) ; (2.14) we obtain for it the expression ( (k T )¡1hv (0)v (¿)i if ¿ > 0; B (¿) = B ® ¯ (2.15) ®¯ 0 if ¿ < 0: Fourier transform of Eq. (2.15) yields Z 1 ¡1 i!t B®¯(!) = (kBT ) dt e hv®(0)v¯(t)i : (2.16) 0 Comparing (2.3) and (2.16), we get the Einstein relation D®¯(!) = kBTB®¯(!): (2.17) Emphasize that the coe±cients B®¯(!) or D®¯(!) contain the full information on the equilibrium state of the system. Besides, the functions B®¯(!) and D®¯(!) are complex, and owing to the causality principle their real and imaginary parts are coupled by the (dispersion) Kramers-Kronig relation. Note also that the energy absorption is proportional to Re B(!). 2.2. LANGEVIN AND FOKKER-PLANCK-KRAMERS EQUATIONS 7 2.2 Langevin and Fokker-Planck-Kramers equations In the ¯rst-order approximation, the role of crystalline substrate reduces to producing of a stationary external potential '(r), in the ¯eld of which an atom moves. When the potential '(r) is one-dimensional, the corresponding Hamiltonian motion equation is always integrable and the motion is regular, hx(t)i / t. However, already for a 2D potential '(r) because of the coupling of the two degrees of freedom the system integrability is destroyed, and in a general case the atom motion becomes stochastic (Lichtenberg), the motion follows the law hx2(t)i / t1+"; t ! 1; (2.18) where the parameter " depends on the shape of potential '(r) as well as on the energy of the atom. Stochastic motion of the adsorbed atom due to coupling of the modes was studied in (ZETF Chirikov), and that due to coupling of the parallel and perpendicular to the surface motion of the adatom, in (my Radiophysics). Besides the adatom's degrees of freedom, there are degrees of freedom corresponding to the motion of substrate atoms. From a general theory of dynamical systems (Lichtenberg) it is known that with increasing of the number of degrees of freedom in a nonlinear system, the chaos becomes more and more \developed", and the character of the motion, more di®usional (i.e. " ! 0 in Eq. (2.18)). However, the complete description of motion of a Hamiltonian system with more than two degrees of freedom is too complicated. Therefore, usually the motion of substrate atoms is treated phenomenologically by introducing a \viscous" friction with a coe±cient ´ (or the tensor ´®¯ in a general case) which is equal to the rate of energy exchange between the adatom and substrate (my SS, Uspechi).